Prove that you have constructed point C on segment EF such that angle ACE is congruent to angle BCF. (Points A and B are on the same side of segment EF, but have different distances to the segment.)
I am not sure if I am on the right track but this is how I have tried prove it: I have proven a new angle, angle ICF, is congruent to angle ACE by the Vertical Angle Theorem (after introducing line AC and point I at the intersect of line AC and line BD (which is perpendicular to segment EF)). I have introduced point J on line BD such that JD=ID and proved triangle CJD is congruent to triangle CID by SAS (CD=CD by Reflexive Property, angle JDC is congruent to angle IDC because line BD is perpendicular to segment EF (which contains point C) therefore angles JDC and IDC are right angles and all right angles are congruent, and I had introduced point J so that JD=ID). At the moment I cannot see how I would prove point J is point B, so I am not sure if the above reasoning is in the right direction.
Thank you for your help!
To prove that you have constructed point C on segment EF such that angle ACE is congruent to angle BCF, you can follow these steps:
1. Draw segment EF, and mark points A and B on the same side of EF but with different distances to the segment.
2. Draw line AC, and construct a perpendicular line BD that intersects AC at point I. This creates right angles at JDC and IDC.
3. Introduce point J on line BD such that JD = ID.
Now, let's break down the proof:
4. Show that angle ACE is congruent to angle ICF: Since angle JDC and angle IDC are right angles, they are congruent. Using the Vertical Angles Theorem, we have angle JDC congruent to angle ICF.
5. Prove triangle CJD congruent to triangle CID: We can use SAS (Side-Angle-Side) congruence to prove the congruence of these triangles.
a. Side CD is congruent to CD (by Reflexive Property).
b. Angle JDC is congruent to angle IDC (both are right angles).
c. Side JD is congruent to ID (by construction).
Therefore, by SAS congruence, we have triangle CJD congruent to triangle CID.
6. Since triangles CJD and CID are congruent, angle CJD is congruent to angle CID. By substitution, angle ACE is congruent to angle ICF, which was proven earlier.
7. Now, to prove that point J is point B, we need to show that JD = IB.
a. JD = ID (by construction).
b. IB = ID (since angle BID is a right angle and triangle BID is a right triangle, the hypotenuse (BD) is equal to the side opposite the right angle (ID)).
Therefore, JD = ID = IB, which implies point J is point B.
8. Finally, since JD = IB and point J is point B, this implies that angle ACE is congruent to angle BCF.
By following these steps, you have proven that you have constructed point C on segment EF such that angle ACE is congruent to angle BCF.